Set Auto-Lock to Never. I think this will help your issue. However, with Never, you must remember to lock your phone manually. And I think the last time I used the download feature I had to keep the app open, without allowing it to background.

The TiVo app developers could - and should - make it so the app doesn't allow the phone to sleep when downloading. They could also make it a setting the user sets to enable/disable. Several of my apps have this feature.

In case a TiVo rep is reading this, ask your development team to read up on this setting: